We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space can be decomposed by means of the reducing subspaces of A. The sum of two functions \( {Q}_i\in {N}_{\kappa_i}\left(\mathcal{H}\right) \), i = 1, 2, minimally represented by the triplets \( \left({\mathcal{K}}_i,{A}_i,{\Gamma}_i\right) \) is also studied. For this purpose, we create a model \( \left(\overset{\sim }{\mathcal{K}},\overset{\sim }{A},\overset{\sim }{\Gamma}\right) \) to represent Q := Q1 + Q2 in terms of \( \left({\mathcal{K}}_i,{A}_i,{\Gamma}_i\right) \). By using this model, necessary and sufficient conditions for κ = κ1+ κ2 are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and the decomposition of the corresponding function Q.
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References
R. Arens, “Operational calculus of linear relations,” Pacific J. Math., 11, 9–23 (1961).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications, New York (1993).
D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo, “Schur functions, operator colligations, and reproducing kernel Pontryagin spaces,” Operator Theory: Advances and Applications, 96, Birkhäuser Verlag, Basel (1997).
J. Behrndt, H. de Snoo, and S. Hassi, “Boundary value problems, Weyl functions, and differential operators,” Monogr. Math., 108 (2020); https://doi.org/10.1007/978-3-030-36714-5.
J. Bognar, Indefinite Inner Product Spaces, Springer-Verlag, New York-Heidelberg (1974).
M. Borogovac, “Inverse of generalized Nevanlinna function that is holomorphic at infinity,” North-West. Europ. J. Math., 6, 19–43 (2020).
M. Borogovac and H. Langer, “A characterization of generalized zeros of negative type of matrix functions of the class \( {N}_{\kappa}^{nxn} \),” Oper. Theory Adv. Appl., 28, 17–26 (1988).
K. Daho and H. Langer, “Matrix functions of the class \( {N}_{\kappa}^{nxn} \),” Math. Nachr., 120, 275–294 (1985).
A. Dijksma, H. Langer, and H. S. V. de Snoo, “Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions,” Math. Nachr., 161, 107–154 (1993).
S. Hassi, H. S. V. de Snoo, and H. Woracek, “Some interpolation problems of Nevanlinna–Pick type,” Oper. Theory Adv. Appl., 106, 201–216 (1998).
I. S. Iohvidov, M. G. Krein, and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Akademie-Verlag, Berlin (1982).
M. G. Kreĭn and H. Langer, “Über die Q-Funktion eines π-hermiteschen Operators im Raume Πκ,” Acta Sci. Math. (Szeged), 34, 191–230 (1973).
M. G. Kreĭn and H. Langer, “Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen,” Math. Nachr., 77, 187–236 (1977).
H. Langer and B. Textorius, “On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space,” Pacific J. Math., 72, No. 1, 135–165 (1977).
A. Luger, “Generalized Nevanlinna functions: operator representations, asymptotic behavior,” Operator Theory (2014), p. 345–371; https://doi.org/10.1007/978-3-0348-0667-1-35.
A. Luger, “A characterization of generalized poles of generalized Nevanlinna functions,” Math. Nachr., 279, 891–910 (2006).
H. de Snoo and H.Woracek, “The Krein formula in almost Pontryagin spaces. A proof via orthogonal coupling,” Indag. Math. (N.S.), 29, No. 2, 714–729 (2018).
P. Sorjonen, “On linear relations in an indefinite inner product space,” Ann. Acad. Sci. Fenn. Ser. A I Math., 4, No. 1, 169–192 (1979).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 7, pp. 893–920, July, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i7.6084.
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Borogovac, M. Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions. Ukr Math J 74, 1021–1052 (2022). https://doi.org/10.1007/s11253-022-02118-x
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DOI: https://doi.org/10.1007/s11253-022-02118-x